Abstract
The purpose of this paper is to introduce a new method based on the deep neural network method (DNN) for finding numerical solution of a novel class of stochastic biological systems. DNN utilizes the Morgan-Voyce even Lucas polynomials and \(\sinh\) function as activation functions of the deep structure. To train this neural network, we utilize the standard Brownian motion, Gauss–Legendre quadrature, and classical optimization algorithm. In the proposed method, acceptable approximate solutions are achieved by employing only a few number of the basis functions. Furthermore, we show convergence of the computational technique. Finally, the numerical technique is implemented for a stochastic biological system to illustrate the effectiveness of the presented strategy.
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References
Aarato M (2003) A famous nonlinear stochastic equation (Lotka–Volterra model with diffusion). Math Comput Model 38:709–726
Baker N, Alexander F, Bremer T, Hagberg A, Kevrekidis Y, Najm H, Parashar M, Patra A, Sethian J, Wild S et al (2019) Workshop report on basic research needs for scientific machine learning: Core technologies for artificial intelligence. Tech. rep, USDOE Office of Science (SC), Washington, DC (United States)
Barikbin MS, Vahidi AR, Damercheli T, Babolian E (2020) An iterative shifted Chebyshev method for nonlinear stochastic Itô-Volterra integral equations. J Comput Appl Math 378:112912
Chakraverty S, Mall S (2017) Artificial neural networks for engineers and scientists: solving ordinary differential equations. CRC Press, Boca Raton
Dareiotis K, Leahy JM (2016) Finite difference schemes for linear stochastic integro-differential equations. Stoch Processes Appl 126(10):3202–3234
Elsken T, Metzen JH, Hutter F et al (2019) Neural architecture search: a survey. J Mach Learn Res 20(55):1–21
Gao J, Liang H, Ma S (2019) Strong convergence of the semi-implicit Euler method for nonlinear stochastic Volterra integral equations with constant delay. Appl Math Comput 348:385–398
Geist M, Petersen Ph, Raslan M, Schneider R, Kutyniok G (2021) Numerical solution of the parametric diffusion equation by deep neural networks. J Sci Comput 88:22
Haber E, Ruthotto L (2017) Stable architectures for deep neural networks. Inverse Prob 34(1):014004
Hadian Rasanan AH, Bajalan N, Parand K, Amani Rad J (2019) Simulation of nonlinear fractional dynamics arising in the modeling of cognitive decision making using a new fractional neural network. Math Meth Appl Sci 43(3):1437–1466
Hajimohammadi Z, Parand K, Ghodsi A (2021) Legendre deep neural network (LDNN) and its application for approximation of nonlinear Volterra–Fredholm–Hammerstein integral equations. ar**v preprint ar**v, 2106, 14320
Han J, Jentzen A, Weinan E (2018) Solving high-dimensional partial differential equations using deep learning. Proc Natl Acad Sci 115(34):8505–8510
Heydari MH, Avazzadeh Z, Mahmoudi MR (2019) Chebyshev cardinal wavelets for nonlinear stochastic differential equations driven with variable-order fractional Brownian motion. Chaos, Solitons Fractals 124:105–124
Heydari MH, Hooshmandasl MR, Barid Loghmani Gh, Cattani C (2016) Wavelets Galerkin method for solving stochastic heat equation. Int J Comput Math 93(9):1579–1596
Heydari MH, Hooshmandasl MR, Cattani C (2020) Wavelets method for solving nonlinear stochastic Itô-Volterra integral equations. Georgian Math J 27(1):81–95
Heydari MH, Hooshmandasl MR, Cattani C, Maalek Ghaini FM (2015) An efficient computational method for solving nonlinear stochastic Itô integral equations: Application for stochastic problems in physics. J Comput Phys 283:148–168
Heydari MH, Hooshmandasl MR, Maalek Ghaini FM, Cattani C (2014) A computational method for solving stochastic Itô-Volterra integral equations based on stochastic operational matrix for generalized hat basis functions. J Comput Phys 270:402–415
Heydari MH, Hooshmandasl MR, Shakiba A, Cattani C (2016) Legendre wavelets Galerkin method for solving nonlinear stochastic integral equations. Nonlinear Dyn 85:1185–1202
Jianyu L, Siwei L, Yingjian Q, Ya** H (2003) Numerical solution of elliptic partial differential equation using radial basis function neural networks. Neural Netw 16:729–734
Kashem BE (2021) A new approach of Morgan-Voyce polynomial to solve three point boundary value problems, IHICPAS
Khodabin M, Maleknejad K, Rostami M, Nouri M (2012) Interpolation solution in generalized stochastic exponential population growth model. Appl Math Model 36(3):1023–1033
Kumar Y, Singh S, Srivastava N, Singh A, Singh VK (2020) Wavelet approximation scheme for distributed order fractional differential equations. Comput Math Appl 80:1985–2017
LeCun Y, Bengio Y, Hinton G (2015) Deep learning. Nature 521(7553):436–444
Li K, Kou J, Zhang W (2019) Deep neural network for unsteady aerodynamic and aeroelastic modeling across multiple Mach numbers. Nonlinear Dyn 96(3):2157–2177
Liu W, Wang Z, Liu X, Zeng N, Liu Y, Alsaadi FE (2017) A survey of deep neural network architectures and their applications. Neurocomputing 234:11–26
Long Z, Lu Y, Ma X, Dong B (2018) Pde-net: Learning pdes from data. In: International conference on machine learning, 3208-3216
Malek A, Beidokhti Shekari R (2006) Numerical solution for high order deferential equations, using a hybrid neural network-Optimization method. Appl Math Comput 183:260–271
Mall S, Chakraverty S (2015) Numerical solution of nonlinear singular initial value problems of Emden-Fowler type using Chebyshev neural network method. Neurocomputing 149:975–982
Masood Z, Majeed K, Samar R, Raja MAZ (2017) Design of Mexican Hat wavelet neural networks for solving Bratu type nonlinear systems. Neurocomputing 221:1–14
Meade AJ Jr, Fernandez AA (1994) The numerical solution of linear ordinary differential equations by feed forward neural networks. Math Comput Model 19:1–25
Meade AJ Jr, Fernandez AA (1994) Solution of nonlinear ordinary differential equations by feedforward neural networks. Math Comput Model 20:19–44
Nouiehed M, Razaviyayn M (2018) Learning deep models: critical points and local openness, arxiv preprint arxiv, 1803.02968
Pakdaman M, Ahmadian A, Effati S, Salahshour S, Baleanu D (2017) Solving differential equations of fractional order using an optimization technique based on training artificial neural network. Appl Math Comput 293:81–95
Peiris V, Sharon N, Sukhorukova N, Ugon J (2021) Generalised rational approximation and its application to improve deep learning classifiers. Appl Math Comput 389:125560
Plate E, Bruti-Liberati N (2010) Numerical solution of stochastic differential equations with jumps in finance. Springer, Berlin, p 64
Rahimkhani P (2023) Numerical solution of nonlinear stochastic differential equations with fractional Brownian motion using fractional-order Genocchi deep neural networks. Commun Nonlinear Sci Numer Simul. https://doi.org/10.1016/j.cnsns.2023.107466
Rahimkhani P, Ordokhani Y (2021) Orthonormal Bernoulli wavelets neural network method and its application in astrophysics. Comput Appl Math 40(30):1–24
Rahimkhani P, Ordokhani Y (2022) Chelyshkov least squares support vector regression for nonlinear stochastic differential equations by variable fractional Brownian motion. Chaos, Solitons Fractals 163:112570
Rahimkhani P, Ordokhani Y (2023) Fractional-order Bernstein wavelets for solving stochastic fractional integro-differential equations. Int J Nonlinear Anal Appl. https://doi.org/10.22075/ijnaa.2022.20273.2141
Rahimkhani P, Ordokhani Y (2023) Performance of Genocchi wavelet neural networks and least squares support vector regression for solving different kinds of differential equations. Comput Appl Math 42:71
Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. J Comput Phys 378:686–707
Raja MAZ, Mehmood J, Sabir Z, Nasab AK, Manzar MA (2019) Numerical solution of doubly singular nonlinear systems using neural networks-based integrated intelligent computing. Neural Comput Appl 31:793–812
Rue P, Villa-Freixa J, Burrage K (2010) Simulation methods with extended stability for stiff biochemical kinetics. BMC Syst Biol 4(1):110
Selvaraju N, Abdul Samant J (2010) Solution of matrix Riccati differential equation for nonlinear singular system using neural networks. Int J Comput Appl 29:48–54
Shannon AG, Horadam AF (1999) Some relationships among Vieta, Morgan-Voyce and Jacobsthal Polynomials. Appl Fibonacci Numbers, pp 307–323
Zheng S, Song Y, Leung T, Goodfellow I (2016) Improving the robustness of deep neural networks via stability training. In: Proceedings of the IEEE conference on computer vision and pattern recognition, pp 4480–4488
Zheng Z, Hong P (2018) Robust detection of adversarial attacks by modeling the intrinsic properties of deep neural networks. In: Proceedings of the 32nd international conference on neural information processing systems, pp 7924–7933
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We express our sincere thanks to the anonymous referees for their valuable suggestions that improved the final manuscript.
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Rahimkhani, P. Deep Neural Network for Solving Stochastic Biological Systems. Iran J Sci 48, 687–696 (2024). https://doi.org/10.1007/s40995-023-01562-z
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DOI: https://doi.org/10.1007/s40995-023-01562-z
Keywords
- Deep neural network
- Stochastic biological systems
- Morgan-Voyce even Lucas polynomials
- Numerical solution